Koopman dynamics of a Hamiltonian flow

Core idea

For a complete Hamiltonian flow $\Phi^t$ with an invariant probability measure and a strongly continuous Koopman group $U_tf=f\circ\Phi^t$ , the generator on smooth observables in its domain is

Lf=X_Hf=\{f,H\}.

The Poisson bracket uses the convention $\iota_{X_H}\omega=dH$ . Thus state evolution $\dot x=X_H(x)$ induces the linear observable equation $\partial_tu=Lu$ .

Densities and conserved quantities

The adjoint evolution is $U_t^*=U_{-t}$ , which transports densities relative to the invariant measure. A conserved observable satisfies $Lf=0$ on the generator domain and is fixed by the flow.

The Hamiltonian itself is conserved. If it is not almost everywhere constant, some bounded function of it is a nonconstant invariant observable, obstructing ergodicity for a probability measure spread across multiple energies. Ergodicity is consequently often studied on a fixed energy surface with its own invariant probability measure.